A dichotomy result for a modified Schr\"odinger equations on unbounded domains

This article aims to investigate the existence of bounded positive solutions of problem $$(P)\qquad \left\{ \begin{array}{ll} - {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = g(x,u) &\hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,} \end{array}\right.$$ with $A_t(x,t,\xi) = \frac{\partial A}{\partial t}(x,t,\xi)$, $a(x,t,\xi) = \nabla_\xi A(x,t,\xi)$ for a given $A(x,t,\xi)$ which grows as $|\xi|^p + |t|^p$ , $p > 1$, where $\Omega \subseteq \mathbb{R}^N$, $N \ge 2$, is an open connected domain with Lipschitz boundary and infinite Lebesgue measure, eventually $\Omega = \mathbb{R}^N$, which generalizes the modified Schr\"odinger equation $$- {\rm div} ((A^*_1(x) + A^*_2(x)|u|^{s}) \nabla u) + \frac{s}2 A^*_2(x)\ |u|^{s - 2} u\ |\nabla u|^2 + u\ =\ |u|^{\mu-2}u \quad\hbox{in $\mathbb{R}^3$.}$$ Under suitable assumptions on $A(x,t,\xi)$ and $g(x,t)$, problem $(P)$ has a variational structure. Then, even in lack of radial symmetry hypotheses, one bounded positive solution of $(P)$ can be found by passing to the limit on a sequence $(u_k)_k$ of bounded solutions on bounded domains. Furthermore, if stronger hypotheses are satisfied, either such a solution is nontrivial or a constant $\bar{\lambda} > 0$ and a sequence of points $(y_k)_k \subset \mathbb{R}^N$ exist such that $$|y_k| \to +\infty\qquad \hbox{and}\qquad \int_{B_1(y_k)} |u_k|^p dx \ge \bar{\lambda}\quad \hbox{for all $k \ge 1$.}$$